It's actually trivial to give a (infinite but) countable set of (non-orthogonal) basis vectors: 2^i for integer i. 0 and 1 are both rationals (scalars), and every real number has a (possibly infinite) binary expansion, eg e = 10.1011011111100001... = 1·2^1 + 0·2^0 + 1·2^-1 + 0·2^-2 + ...
What I think you want to say is that "any real number has a binary expansion". Which is true, but the binary sequences don't form a vector space over R, but a Z/2-module. And as a Z/2-module, your { 2^i for integer i } isn't even a basis because you need infinite expansions to express most real numbers. The span of a basis are only the finite linear combinations.
(FWIW, I think you've given a description of the dyadic rationals.)